Study the analogy between force and torque, mass and moment of inertia, and linear acceleration and angular acceleration.

If you have ever spun a bike wheel or pushed a merry-go-round, you know that force is needed to change angular velocity as seen in
[link] . In fact, your intuition is reliable in predicting many of the factors that are involved. For example, we know that a door opens slowly if we push too close to its hinges. Furthermore, we know that the more massive the door, the more slowly it opens. The first example implies that the farther the force is applied from the pivot, the greater the angular acceleration; another implication is that angular acceleration is inversely proportional to mass. These relationships should seem very similar to the familiar relationships among force, mass, and acceleration embodied in Newton’s second law of motion. There are, in fact, precise rotational analogs to both force and mass.

Force is required to spin the bike wheel. The greater the force, the greater the angular acceleration produced. The more massive the wheel, the smaller the angular acceleration. If you push on a spoke closer to the axle, the angular acceleration will be smaller.

To develop the precise relationship among force, mass, radius, and angular acceleration, consider what happens if we exert a force
Fsize 12{F} {} on a point mass
msize 12{m} {} that is at a distance
rsize 12{r} {} from a pivot point, as shown in
[link] . Because the force is perpendicular to
rsize 12{r} {} , an acceleration
a=Fmsize 12{a= { {F} over {m} } } {} is obtained in the direction of
Fsize 12{F} {} . We can rearrange this equation such that
F=masize 12{F= ital "ma"} {} and then look for ways to relate this expression to expressions for rotational quantities. We note that
a=rαsize 12{a=rα} {} , and we substitute this expression into
F=masize 12{F= ital "ma"} {} , yielding

F=mrα.size 12{F= ital "mr"α"."} {}

Recall that
torque is the turning effectiveness of a force. In this case, because
Fsize 12{"F"} {} is perpendicular to
rsize 12{r} {} , torque is simply
τ=Frsize 12{τ=rα} {} . So, if we multiply both sides of the equation above by
rsize 12{r} {} , we get torque on the left-hand side. That is,

rF=mr2αsize 12{ ital "rF"= ital "mr" rSup { size 8{2} } α} {}

or

τ=mr2α.size 12{τ= ital "mr" rSup { size 8{2} } α.} {}

This last equation is the rotational analog of Newton’s second law (
F=masize 12{F= ital "ma"} {} ), where torque is analogous to force, angular acceleration is analogous to translational acceleration, and
mr2size 12{ ital "mr" rSup { size 8{2} } } {} is analogous to mass (or inertia). The quantity
mr2size 12{ ital "mr" rSup { size 8{2} } } {} is called the
rotational inertia or
moment of inertia of a point mass
msize 12{m} {} a distance
rsize 12{r} {} from the center of rotation.

An object is supported by a horizontal frictionless table and is attached to a pivot point by a cord that supplies centripetal force. A force
Fsize 12{F} {} is applied to the object perpendicular to the radius
rsize 12{r} {} , causing it to accelerate about the pivot point. The force is kept perpendicular to
rsize 12{r} {} .

Making connections: rotational motion dynamics

Dynamics for rotational motion is completely analogous to linear or translational dynamics. Dynamics is concerned with force and mass and their effects on motion. For rotational motion, we will find direct analogs to force and mass that behave just as we would expect from our earlier experiences.

Questions & Answers

find the density of a fluid in which a hydrometer having a density of 0.750g/mL floats with 92.0% of its volume submerged.

(a)calculate the buoyant force on a 2.00-L Helium balloon.(b) given the mass of the rubber in the balloon is 1.50g. what is the vertical force on the balloon if it is let go? you can neglect the volume of the rubber.

A disturbance that travel from one medium to another and without causing permanent change to its displacement

Fagbenro

In physics, a wave is a disturbance that transfers energy through matter or space, with little or no associated mass transport (Mass transfer). ... There are two main types ofwaves: mechanical and electromagnetic. Mechanicalwaves propagate through a physical matter, whose substance is being deformed

Devansh

K

Manyo

thanks jare

Doc

Thanks

AMADI

Note:
LINEAR MOMENTUM
Linear momentum is defined as the product of a system’s mass multiplied by its velocity:
size 12{p=mv} {}